# The impact of long-range dispersal on gene surfing

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Edited by Marcus W. Feldman, Stanford University, Stanford, CA, and approved February 21, 2020 (received for review November 8, 2019)

## Significance

The expansion of a population into a new area occurs frequently in nature. Examples include epidemic outbreaks, invasions by nonnative species, and colonization of regions rendered hospitable by climate change. Many expansions are driven by long-range migration of organisms or seeds borne by air, water, or traveling hosts. These events amplify the impact of random chance on genetic variation: A lucky individual which reaches unoccupied territory first can have more descendants than subsequent arrivals, even without advantageous mutations. We model the patterns of variation generated by random long-range migration events and study their effects on genetic diversity. Our results elucidate the evolutionary consequences of long-range dispersal and are important for correctly interpreting sources of variation in geographically structured genomic data.

## Abstract

Range expansions lead to distinctive patterns of genetic variation in populations, even in the absence of selection. These patterns and their genetic consequences have been well studied for populations advancing through successive short-ranged migration events. However, most populations harbor some degree of long-range dispersal, experiencing rare yet consequential migration events over arbitrarily long distances. Although dispersal is known to strongly affect spatial genetic structure during range expansions, the resulting patterns and their impact on neutral diversity remain poorly understood. Here, we systematically study the consequences of long-range dispersal on patterns of neutral variation during range expansion in a class of dispersal models which spans the extremes of local (effectively short-ranged) and global (effectively well-mixed) migration. We find that sufficiently long-ranged dispersal leaves behind a mosaic of monoallelic patches, whose number and size are highly sensitive to the distribution of dispersal distances. We develop a coarse-grained model which connects statistical features of these spatial patterns to the evolution of neutral diversity during the range expansion. We show that growth mechanisms that appear qualitatively similar can engender vastly different outcomes for diversity: Depending on the tail of the dispersal distance distribution, diversity can be either preserved (i.e., many variants survive) or lost (i.e., one variant dominates) at long times. Our results highlight the impact of spatial and migratory structure on genetic variation during processes as varied as range expansions, species invasions, epidemics, and the spread of beneficial mutations in established populations.

Range expansions have occurred in the history of many species, from plants (1) to avian (2), aquatic (3, 4), and terrestrial (5, 6) animals, including humans (7). Over geological timescales, they have been driven by climactic changes such as glacial advance and melting in the northern hemisphere (8, 9). More recently, anthropogenic climate change and human-mediated introduction of invasive species have driven the expansion of species into new territory (10). These expansion events impact the genetic makeup of the population, in ways that are dramatically different from population expansions without spatial structure (11). In particular, neutral mutations occurring during range expansions leave behind signatures that are otherwise associated with selection, such as sweeps through the population (12), allelic gradients (13), and reduction in local genetic diversity (14⇓–16). Understanding the patterns of neutral variation left behind by range expansions is crucial for disentangling the role of spatial structure from selection in determining genetic diversity (17).

Much of our current understanding of neutral evolution during range expansions is derived from situations where individuals migrate a short distance between generations (12⇓⇓⇓–16, 18). In this case, the population advances through a wave of roughly constant speed separating occupied and unoccupied regions of space. Crucially, only individuals that happen to be close to the advancing front contribute to future generations, and large swaths of the population after the expansion can be traced back to a few individuals at the edge of the originating population, a phenomenon termed gene surfing. The resulting neutral variation shows a characteristic pattern: Local diversity is strongly reduced as neutral variants segregate into uniform regions, called sectors, in which a single allele dominates. Nevertheless, in a radial expansion, sectors of different variants persist at long times as a result of which global diversity is maintained, as seen in Fig. 1 *A*–*D*. These patterns persist under moderate levels of gene flow due to subsequent diffusion following the initial advance (18, 19).

Most organisms, however, experience some amount of long-range dispersal (20, 21). Pollen, seeds, and microorganisms are dispersed over long distances by wind and water or by wandering or migratory animals whose excursions influence their own evolution as well. The resulting distributions of dispersal distances, also called dispersal kernels, are often “fat tailed”: They do not have a characteristic cutoff length scale and fall off slower than exponentially with distance. Empirical measurements of dispersal kernels have shown that fat-tailed kernels arise in the spreading behavior of numerous species (22). Theoretical analyses have established that fat-tailed dispersal kernels accelerate expansion dynamics (20, 23), allowing the size of the expanding population to grow faster than linearly with time, and strongly influence population structure by breaking up the wave of advance associated with short-ranged spreading (24⇓⇓⇓⇓–29).

Although long-range dispersal is recognized as being consequential for range expansions (11, 30⇓–32), its precise effects on genetic diversity are not fully understood. Whereas it is recognized that dispersal leads to monoallelic patches (24⇓⇓–27), the conditions for patches to dominate over sectors have not been identified. Furthermore, the structural characteristics and dynamics of the patch patterns and their subsequent impact on neutral diversity have not been systematically studied. As a result, even the basic question of whether neutral variation in the originating population is maintained during dispersal-driven range expansions is unresolved. Simulation studies involving mixtures of two Gaussian (i.e., non–fat-tailed) dispersal kernels with different mean distances have shown support for contrasting effects of increasing the weight of the broader dispersal kernel on neutral diversity (33) (the so-called “embolism effect”). At low levels, founder events ahead of the expanding front of the population wipe out diversity, but at higher levels, diversity in the expanding region is maintained by serial reintroduction of variants from the interior of the population. However, other studies have argued that a reduction in diversity due to the embolism effect occurs only for dispersal along corridors and for thin-tailed dispersal kernels (34), whereas fat-tailed kernels ought to generically enhance genetic diversity (30, 34⇓⇓–37).

Here, we study the evolution of neutral diversity in a simplified model of range expansions with dispersal events drawn from fat-tailed kernels. By analyzing a class of dispersal kernels which spans the two extremes of well-mixed growth and wavelike spreading, we obtain a comprehensive picture of neutral evolution in dispersal-accelerated range expansions. We find that long-range dispersal breaks up radial sectors into monoallelic patches, or blobs, but only if the kernel is sufficiently fat tailed (Fig. 1). The characteristic size of these blobs relative to the overall size of the population can vary widely, reflecting qualitative differences in the growth dynamics for different dispersal kernels. For the broadest kernels, the spatial distribution of alleles approaches a highly fragmented speckle pattern (Fig. 1 *I*–*L*). These patterns depart strongly from the prevailing paradigm of sectors as the spatial signature of range expansions (11, 14, 18).

We also investigate how global diversity is impacted by the breakup of sectors into blobs and speckles. By studying the growth of the typical number and size of blobs as the range expansion progresses, we show that fat-tailed kernels display the entire range of possible outcomes for neutral variation: Depending on the exponent characterizing the tail of the kernel, the initial diversity can be almost perfectly preserved, or completely lost, as the expansion progresses. Strikingly, we find that long-range dispersal can in some cases erode genetic diversity compared to short-range dispersal, through a mechanism that differs fundamentally from the previously documented embolism effect.

## Model

We consider growth into isotropic space from a compact initial population of size

Before describing the results of multiallele simulations, we briefly summarize the known behavior of range expansions in power-law growth kernels for a monoallelic population (38) (Table 1). The value

## Results

### Long-Range Dispersal Breaks Up Sectors into Monoallelic Blobs.

Fig. 1 shows snapshots from simulations of range expansions for three different kernel exponents. The narrowest kernel, *A*–*C*. At early times, boundaries between sectors of different alleles can annihilate due to the random wandering of sector boundaries from straight radial rays, but at later times, established sectors are stable against annihilation and the allelic fractions become essentially frozen in time (19, 40), up to fluctuations due to the random wandering of the boundaries between sectors. These dynamics are reflected in the evolution of allelic fractions of individual simulations, which settle to a near-constant value at long times (Fig. 1*D*).

When *E*–*G*). Alleles still segregate into monoallelic regions, but these do not form radial sectors. Instead, the population is composed of a mosaic of blobs of varying size, each of which has an irregular boundary but is roughly isotropic in shape. At each time point, a core region surrounding the homeland can be identified within which all sites are occupied. Also visible are isolated clusters of occupied sites, separated from the core by empty sites, which were colonized by the offspring of a single migrant that landed far from the bulk of the population at earlier times. These satellite outbreaks, visible at the outer edges of the population in each snapshot, are a characteristic feature of jump-driven growth in the presence of long-range dispersal (38, 41). The patchiness of jump-driven range expansions arises from the continued accumulation of satellite outbreaks. The resulting disruption of sectors when *Discussion*.

Upon comparing patterns for the two kernels displaying blobs in Fig. 1, the patterns at the broadest kernel, *I*–*K*), show finer and more numerous blobs compared to the intermediate kernel with *E*–*G*). Relative to the population size, blobs also appear to get finer as the range expansion progresses for *H* and *L*) shows different characteristics for the two kernels: Each trajectory exhibits stronger variations over time for the intermediate kernel,

### A Hierarchy of Doublings in Time Determines the Characteristic Size of Blobs.

Although the early establishment of sectors in constant-speed range expansions (such as in Fig. 1 *A*–*C*) is stochastic, their subsequent growth over time is tied to the radial population growth and is essentially deterministic, up to random wandering of sector boundaries which becomes insignificant at long times (the growth in transverse fluctuations of the boundaries is overcome by the linear expansion of the circumference with time). By contrast, the placement and size of the monoallelic regions in the jump-driven range expansions (Fig. 1 *E*–*K*; see Fig. 2*A* for complete time evolution of a 1D simulation) are stochastic at all stages of growth. However, the characteristic sizes of blobs incorporated into the growing core at different times (or equivalently, at different radial distances) follow a distinct pattern in the vicinity of the marginal point *B*). As a result, when the growth of *C*.

The hierarchy connecting core size scales at

To quantitatively test the hierarchy of blob-core sizes predicted above, we measured spatial relationships among pairs of individuals and their most recent common ancestor (MRCA). As Fig. 2*D* illustrates, the time-doubling hierarchy predicts that a pair of individuals centered at distance r are likely to belong to the same satellite if their separation is much less than

Measurements of pair–MRCA relationships in 1D simulations confirm these expectations (Fig. 3; see *Materials and Methods* for details). In all growth regimes, the average MRCA position *A*–*C*). Upon rescaling the pair separation with the proposed blob scale *D*), with a sharp fall in mean MRCA distance when *A* and *B*, *Insets*), since the characteristic blob size falls relative to the size of the entire population as the expansion proceeds. By contrast, for power-law growth the ratio *C*, *Inset*).

### A Coarse-Grained Model of Blob Replication Predicts Distinct Outcomes for Neutral Diversity in Different Growth Regimes.

Next, we investigate the effect of the time-doubling hierarchy of blob sizes on the global neutral genetic diversity as the range expansion progresses. Since each satellite originated from a single founder, the isolated growth of satellites acts as a coarsening mechanism which locally reduces diversity. However, all individuals in the core can contribute long-distance migration events, so the seeding of new satellites provides a mechanism to maintain global genetic diversity in the population. The competition between coarsening and diversification determines the fate of neutral diversity during jump-driven growth.

We now develop a semideterministic model for the evolution of the average neutral heterozygosity in jump-driven range expansions, which combines the deterministic placement and growth of satellite domains with random draws of the allelic identity of each domain. The jump-driven growth dynamic has two consequences for the dynamics of seeding and coarsening: 1) The allelic identity of a typical satellite domain joining the core at time t is determined by a seed drawn from the gene pool of the core at time *SI Appendix*, section 1).

Analytical progress can be made through additional simplifying assumptions. First, we ignore the spatial structure of the core between doublings. Instead, we treat the core at each doubling as a spatially homogeneous mix of alleles from which seeds are randomly drawn. Second, rather than attempting to capture the time evolution of all alleles, we track the global heterozygosity H, a commonly used metric of population-level genetic variation. The heterozygosity is defined as the probability that a random pair sampled from the population will have alleles of different identity. To be concrete, we consider a doubling which evolves a homogeneous core of radial size *D*). The allelic identities of the new satellites are stochastically determined by random draws from the population within *SI Appendix*, section 1 for more details), the expected heterozygosity after one doubling is

Eq. **1** captures the balance between coarsening and seeding in the deterministic approximation to jump-driven growth, with the influence of the growth dynamic manifested in the factor **1** with the evolution of heterozygosity

To connect the evolution described above to the growth of a population out of a spatially homogeneous, compact homeland of population size

We now investigate the evolution of the average heterozygosity predicted by Eq. **2** for the asymptotic growth forms of Table 1. For kernels with **2** to a constant value (see *SI Appendix*, section 1 for details). The coarse-grained model therefore predicts that neutral diversity can be preserved at long times in this model: Complete fixation of one allele is avoided at long times, and a finite average heterozygosity is reached within a few doublings.

In contrast to the stretched-exponential growth rule, a power-law growth rule **2** then predicts that the heterozygosity

Kernels with **2**, to converge to a finite value which lies between *SI Appendix*, section 2). In contrast with the power-law growth regime, the growth of the effective population size allows neutral diversity to be partially preserved over long times; however, the growth of

In summary, the coarse-grained, semideterministic approximation of the jump-driven growth provides a minimal model that allows us to evaluate the competing effects of coarsening and diversification during jump-driven range expansions. The model predicts that when

### Heterozygosity Evolution in Simulations Is Consistent with the Coarse-Grained Model.

To test whether the predictions of the coarse-grained model hold under the full dynamics, we investigate the evolution of neutral diversity in simulations with

For both planar and linear range expansions, the measured average heterozygosities are consistent with the predicted trends. In 2D simulations (Fig. 4*A*), the average heterozygosity does not change significantly between final population sizes of *A*, *Inset* and *SI Appendix*, Fig. A7 for evolution of heterozygosities with population size). However, in the range of kernel exponents *B* and *SI Appendix*, Fig. A8) are qualitatively similar, but with persistent loss of diversity occurring for kernel exponent values

Although the qualitative difference between kernels with **2** occurs over many time-doubling generations. In real time, this implies that the population must grow over many orders of magnitude before the average heterozygosity converges to its limiting value. In *SI Appendix*, Fig. A3, we show that the heterozygosity decay observed in simulations is consistent with Eq. **2**, for several homeland sizes in both 1D and 2D. According to our theory, the largest simulations we have run correspond to roughly 7 time-doubling generations, whereas convergence is expected after 15 to 20 doublings. Therefore, our simulated population sizes are still many orders of magnitude too small for convergence to to be observed.

*SI Appendix*, section 2 also reports additional predictions for the average heterozygosity from the coarse-grained model in the stretched-exponential (*SI Appendix*, Figs. A2 and A4) demonstrate that the coarse-grained model, despite its simplifications and approximations, reproduces many features of the heterozygosity trends for kernels across different regimes.

## Discussion

We have demonstrated that long-range dispersal can dramatically impact the local structures and global trends of neutral genetic diversity left behind by a range expansion. Specifically, dispersal kernels with a power-law tail characterized by exponent *I*–*K*).

As a consequence of a time-doubling hierarchy inherent to jump-driven growth, we have identified an effective population size of blobs generated during a doubling in time. The evolution of this effective population over “time-doubling generations” (which are constant increments in log time) follows distinct trends in the two regimes of jump-driven growth: The effective population size is stagnant in the first growth regime (*SI Appendix*, section 3 and Fig. A6).

The breakup of sectors into blobs and speckles has important genetic consequences. As can be seen in Fig. 1 *A*–*C*, the sector geometry generated by constant-speed range expansions leaves a signature of the direction of expansion on local genetic patterns (14, 18), which can be detected in principal-component maps of genetic variation (42). By contrast, the relation between local blob geometry and global population history in jump-driven expansions (Fig. 1 *E*–*K*) is more subtle: Individual blob shapes are isotropic and do not directly reveal the expansion direction. Blobs do get larger with increasing distance from the homeland on average, but the significant stochastic variation in blob sizes makes this signature difficult to detect. Since neighboring blobs could have been seeded by migrants from well-separated regions, jump-driven range expansions are also expected to have significantly higher levels of mixing compared to sectored expansions. This mixing reduces the positional advantage conferred to mutations which arise near the edge of the expanding population, thus mitigating the gene surfing effect (12, 13). The finer structure and increased mixing due to blobs are likely to have an impact on the evolutionary effects of geographic structure such as reduced adaptive potential (43), response to inbreeding depression (44), and expansion load (45).

Whereas we have focused on describing the expansion of a population into previously unoccupied territory, our results are applicable to other biological expansions as well. The growth dynamics of our model also apply to the spread of beneficial mutations from a localized region into an established wild-type population with a spatially uniform population density (19). In this context, our model would describe the patterns of variation caused by the spread of distinct beneficial mutations with similar fitness effects—a soft sweep (41, 46, 47)—out of a small region (the homeland) which experienced a selection pressure earlier than the rest of the population. Analogues of the patterns and mechanisms described here could also play a role in within-host viral dynamics during infections (48, 49) and in cancer metastasis (50, 51).

An outstanding question regarding the population-genetic consequences of dispersal on range expansions has been whether enhanced dispersal preserves or erodes diversity relative to short-range migration. In a previous study of mixtures of two non–fat-tailed kernels with different characteristic jump lengths, intermediate levels of longer-range jumps were shown to reduce neutral diversity relative to low or high levels in narrow corridors (33). The reduction occurred when a fortuitously placed satellite from a pioneer seed quickly filled the width of the corridor and blocked other alleles from advancing, a mechanism termed the embolism effect [this is a version of the more generic “founder takes all” mechanism (52)]. However, restriction of growth along a narrow corridor is essential for the embolism effect to wipe out genetic variation (34); in a radial expansion, embolisms would suppress diversity only within certain angular ranges, and neutral variation would persist in the form of sectors at long times. Furthermore, the embolism effect was demonstrated for kernels with a strict upper limit to the allowed jump distances, which restricted the ability of individuals from the interior of the population to contribute to diversification. For these reasons, other studies have speculated that fat-tailed kernels without an upper cutoff in dispersal distance might not experience the embolism effect and might enhance diversity relative to short-ranged dispersal in all cases (34⇓⇓–37).

In the context of these previous studies, a key result of our work is that even fat-tailed kernels without a cutoff can induce a loss of neutral variation. Similarly to ref. 33, we have shown that boosting long-range dispersal has a nonmonotonic effect on diversity: Intermediate kernels (i.e., with power-law exponent

Theoretical and computational studies (12, 13, 15, 33⇓⇓⇓–37) and laboratory experiments on model organisms (14), although highly simplified relative to real-world populations, have nevertheless provided heuristics and mechanisms that have guided the interpretation of field genomic data. In the context of range expansions, the concepts of gene surfing, sectors, and embolisms have been invoked to explain patterns of genetic variation in a wide range of plant (53⇓–55), animal (56, 57), and microbial (58) populations. Our work shows that these heuristics are incomplete when long-range dispersal is present: Sectors give way to blobs and speckles with increasing levels of dispersal; the mechanism of engulfment can engender diversity loss even for fat-tailed kernels in the absence of corridors, where the embolism effect does not apply.

Many interesting avenues for further study can be identified. By filling up demes instantaneously and irreversibly, we have focused in this work on the genetic patterns seen immediately after colonization. We have ignored subsequent reshuffling of alleles among demes, which would blur the boundaries between blobs at later times and smear out the predicted spatial patterns over time. This smearing would impact the ability to detect the patchiness of jump-driven range expansions at later times, with the problem being more severe closer to the homeland. While previous studies have shown that gene segregation due to the range expansion persists for appreciable periods of time beyond the colonization for both short-range (18) and long-range (34) dispersal, a more refined spatial model incorporating the exchange of individuals among demes of finite population size would address the question of how long the patterns remain measurable at different distances from the homeland. (Note that the blurring of patterns near the interior during the range expansion does not impact the evolution of global diversity, since the spatial organization of alleles in the core is irrelevant to the identity of satellites generated in the time-doubling hierarchy.) Such a model could also be used to study the interplay of genetic drift within demes and the large-scale diversity evolution captured in our coarse-grained model. In addition, the applicability of our results to continuous populations without a deme structure could be studied by introducing long-range dispersal into continuum population genetics models (59, 60) and simulations (61).

Our results show that spatial constraints fundamentally alter the mathematical structures underlying neutral evolution in expanding populations. Whereas well-mixed populations map onto Markovian Pólya–Eggenberger urn models (62), the hierarchy of doublings in time eliminates Markovianness in the full jump-driven growth: The genetic makeup of individuals entering the population at time t depends on the state of the population at times of order

The coarse-grained model of blob evolution provides a route to understanding the genealogical structures left behind by jump-driven expansion, which is crucial for demographic inference. The effective population of satellites, over generations corresponding to doublings in time, is much simpler to describe compared to the full stochastic dynamics; analyzing the genealogical structure of this effective population in different growth regimes would be a useful first step to understanding genealogies in the full stochastic model. For instance, in the power-law growth regime (

## Materials and Methods

Simulations were implemented in the C++ programming language. Pseudorandom numbers were generated using the Mersenne Twister engine provided in the C++ standard library. Deme positions are quantized to an integer lattice in d dimensions. The simulation keeps track of all occupied demes and the allelic identity (0 or 1) of each deme. To avoid finite-size effects without initializing enormous arrays of mostly empty demes, occupied demes were stored in unordered containers implemented using hash tables (66, 67) (specifically, unordered_map from the C++ Standard Template Library was used). Using this approach, the effective lattice size is

Simulations are initialized by randomly assigning allelic identities to a compact zone around the origin. In 1D, demes are filled in the range

The output of the simulations varied based on the measured quantity. For tracking the heterozygosity, it was sufficient to record the allele fractions at successive population sizes. Images of simulated populations required occupied deme positions and allelic identities to be recorded. Measurements of the MRCA positions were conducted in separate simulations in which the outbreak was begun from a single occupied deme, and a genealogical tree was maintained and updated at each successful jump. At the end of each simulation, the MRCA was recorded for pairs positioned at different center–pair distances from the origin.

Run times are determined by the number of failed attempts made as the simulation progresses toward a target population size. The run time for individual simulations ranged from a few minutes to 72 h and was significantly higher for kernels with

### Code Availability.

Code used to generate simulation data will be made available by the authors upon request.

## Acknowledgments

Research reported in this publication was supported by an NSF CAREER award (1555330) and by a Simons Investigator award from the Simons Foundation (327934). This research used resources of the National Energy Research Scientific Computing Center, a US Department of Energy Office of Science User Facility operated under Contract DE-AC02-05CH11231. This work benefited from access to the University of Oregon high-performance computer, Talapas.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: ohallats{at}berkeley.edu.

Author contributions: J.P. and O.H. designed research, performed research, and wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1919485117/-/DCSupplemental.

Published under the PNAS license.

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